London have no frontiers. If you want, you can put them (the river, for example). But I don´t recommended it !!!
It´s London too !!! The taxi are black (well, … a “little” black, and colorful), the squirrel are jumping and eating chestnuts, … We are in London !!! But … Where are the Big Ben, The Parliament Houses, Chelsea, Piccadilly, …The Queen !!! All of them are in the “other side”. I may say “in the wild side”
The Southbank is more quietly, more “analogue”. I love to walk along and across this part of London. You can take photos, and when you show it to your mother, she will say: Are you sure you go to the right town? Is this London?
You have great places, like the London Eye (Is it more fashion & chic the “little” Piccadilly Circus? Trafalgar Square?), The Tate modern (Have you ever seen the “old” Tate? What do you think? What is the most beautiful?).
And you can see the city, the “old town” … besides the river, away from the noise, with a different perspective.